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Sharp threshold for universality of cokernels of classical random matrix models over the $p$-adic integers

Jiwan Jung, Jungin Lee, Myungjun Yu

Abstract

We prove that $\frac{\log n}{n}$ is the sharp threshold for universality of the distribution of cokernels of random matrices over $\mathbb{Z}_p$. More precisely, let $α_n = \frac{c\log n}{n}$ for a constant $c>0$ and let $A(n)$ be an $α_n$-balanced random matrix over $\mathbb{Z}_p$. For non-symmetric, symmetric, and alternating matrix models, we prove that if $c>1$, then the limiting distribution of the cokernel of $A(n)$ coincides with the universal distribution of the corresponding symmetry type, whereas universality fails at the critical scale $c=1$. This improves earlier universality results, which required $α_n \gg \frac{\log n}{n}$, to the optimal threshold. As an application, we generalize the universality result for Sylow $p$-subgroups of sandpile groups of Erdős-Rényi random graphs to a broader class of Erdős-Rényi graph sequences. Our approach is based on a unified framework that simultaneously treats all symmetry types of random matrices as well as the random graph model, rather than handling each case separately.

Sharp threshold for universality of cokernels of classical random matrix models over the $p$-adic integers

Abstract

We prove that is the sharp threshold for universality of the distribution of cokernels of random matrices over . More precisely, let for a constant and let be an -balanced random matrix over . For non-symmetric, symmetric, and alternating matrix models, we prove that if , then the limiting distribution of the cokernel of coincides with the universal distribution of the corresponding symmetry type, whereas universality fails at the critical scale . This improves earlier universality results, which required , to the optimal threshold. As an application, we generalize the universality result for Sylow -subgroups of sandpile groups of Erdős-Rényi random graphs to a broader class of Erdős-Rényi graph sequences. Our approach is based on a unified framework that simultaneously treats all symmetry types of random matrices as well as the random graph model, rather than handling each case separately.
Paper Structure (14 sections, 29 theorems, 115 equations)

This paper contains 14 sections, 29 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. The moment
  3. Setting
  4. Moment formula for each case
  5. A general formula
  6. Main term of the moments
  7. Bilinear pairings and maximal isotropic subgroups
  8. Error term of the moments
  9. Some lemmas
  10. Proof of Theorem \ref{['thm: general']}: Basic setting
  11. Proof of Theorem \ref{['thm: general']}: Cases C1 and C2
  12. Proof of Theorem \ref{['thm: general']}: Case C3
  13. Sharpness of the aln threshold
  14. Extending to a finite set of primes

Key Result

Theorem 1.1

(NW22) Let $(\alpha_n)_{n \ge 1}$ be a sequence of positive real numbers in $(0,1/2]$ such that for every constant $\Delta > 0$, we have $\alpha_n \geq \frac{\Delta \log n}{n}$ for all sufficiently large $n$. Let $A(n)$ be an $\alpha_n$-balanced random matrix in $\mathrm{M}_n(\mathbb{Z}_p)$ for each

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 37 more